Remark 2 From the first item of the above definition it follows that the homology classes of \(Z_{1},Z_{2}\)  in \(H_*(X,\mathbb {Z})\)  are the same, see for instance [1]} for this implication. It is natural to expect that the second item holds automatically, however, this does not seem to be the case. The proof in the mentioned reference is not adaptable to provide this stronger statement.

